Next Article in Journal
Ulam Stability of Second-Order Periodic Linear Differential Equations with Periodic Damping
Previous Article in Journal
Enhanced Oscillation Criteria for Non-Canonical Second-Order Advanced Dynamic Equations on Time Scales
 
 
Font Type:
Arial Georgia Verdana
Font Size:
Aa Aa Aa
Line Spacing:
Column Width:
Background:
Article

Gröbner–Shirshov Bases for Temperley–Lieb Algebras of Type F

1
Department of Mathematics Education, Catholic Kwandong University, Gangneung-si 25601, Republic of Korea
2
Department of Mathematics, Seoul Women’s University, Seoul 01797, Republic of Korea
*
Author to whom correspondence should be addressed.
Symmetry 2024, 16(11), 1458; https://doi.org/10.3390/sym16111458
Submission received: 9 August 2024 / Revised: 5 October 2024 / Accepted: 31 October 2024 / Published: 3 November 2024
(This article belongs to the Section Mathematics)

Abstract

:
For the Temperley–Lieb algebras of type F n with n 4 , we construct their Gröbner–Shirshov bases. Explicitly, the corresponding finite sets consisting of the standard monomials of type F n , which are exactly the fully commutative elements of F n , are enumerated when n = 4, 5, and 6.
MSC:
Primary 16Z10; Secondary 05E16; 20F55; 68W30

1. Introduction

The Temperley–Lieb algebra stems from the context of statistical mechanics in the 1970s [1]. Later, its structure was studied in connection with knot theory, where it is known to be a quotient of the Hecke algebra of type A in [2].
Our main tool for understanding the structure of the Temperley–Lieb algebra is from the Gröbner–Shirshov basis theory, that is, the noncommutative Gröbner basis theory, which provides a powerful method for finding the structure of (non-)associative algebras and their representations, especially in computational aspects. The central interest in the notion of Gröbner–Shirshov bases originates from Buchberger’s algorithm for computing Gröbner bases for commutative algebras [3] and independently from Shirshov’s Composition Lemma and his algorithm for Lie algebras [4,5]. After a decade, Bokut applied Shirshov’s method to associative algebras [6], and Bergman mentioned the diamond lemma for ring theory [7]. The main strategy of the composition–diamond lemma is to establish an algorithm for constructing standard monomials of a quotient algebra by a two-sided ideal generated by a set of relations called a Gröbner–Shirshov basis.
One of the applications of Gröbner–Shirshov bases in group theory is the word problem [8]. We see that it is solvable in a finite Coxeter group since any element has a unique normal form with respect to a Gröbner–Shirshov basis. It is worthwhile to mention that a Coxeter group is the group of symmetries of a regular or semiregular polytope in the Euclidean space. Some computational results on Coxeter groups appeared in a series of papers [9,10,11,12,13,14,15]. In addition, Gröbner–Shirshov bases for various structures were investigated by the Bokut school in Guangzhou [16]. It is notable to mention that there are algebras with finite Gröbner–Shirshov bases and unsolvable zero divisors problems [17], and there are also some examples from different algebraic structures [18].
For any Temperley–Lieb algebra, its basis consisting of standard monomials coming from a Gröbner–Shirshov basis is indexed by a subset of fully commutative elements [19,20]. The fully commutative element of a Coxeter group is an element with no braid relation in its reduced expression. Motivated by the bijective correspondence of the homogeneous representations of KLR algebras (or quiver Hecke algebras) with fully commutative elements of the corresponding Coxeter group (for simply laced cases only) [21], Feinberg and Lee studied the packet decomposition of fully commutative elements for the Coxeter groups of types A and D [22]. It is known that the finite sets of fully commutative elements for the Coxeter groups are classified into seven families: A n , B n , D n , E n , F n , H n ,   and   I 2 ( m ) [19,23].
For Temperley–Lieb algebras of types A, B, and D (moreover, of imprimitive complex reflection groups), their explicit basis elements are enumerated in the papers [9,24,25]. In this article, we focus on the algebras of type F.

2. Gröbner–Shirshov Bases

Let X be a set and let X be the free monoid of associative words on X. We denote the empty word by 1 and the length (or degree) of a word u by l ( u ) . We also define a monomial order < on X to be a well ordered such that x < y implies a x b < a y b for all a , b X .
We fix a monomial order < on X and let F X be the free associative algebra generated by X over a field F . Given a nonzero element p F X , we denote by p ¯ the maximal monomial (called the leading monomial) appearing in p under the ordering <. Thus
p = α p ¯ + β i w i
with α , β i F , w i X , α 0 , and w i < p ¯ . If α = 1 , p is said to be monic.
Let S be a subset of monic elements in F X , and let I be the two-sided ideal of F X generated by S. Then, we say that the algebra A = F X / I is defined by S.
Definition 1. 
Given a subset S of monic elements in F X , a monomial u X is said to be S-standard (or S-reduced) if u a s ¯ b for any s S and a , b X . Otherwise, the monomial u is said to be S-reducible.
Lemma 1 
([6,7]). Every p F X can be expressed as
p = α i a i s i b i + β j u j ,
where α i , β j F , a i , b i , u j X , s i S , a i s i ¯ b i p ¯ , u j p ¯ , and u j are S-standard.
Remark 1. 
The term β j u j in the expression (1) is called a normal form (or a remainder) of p with respect to the subset S (and with respect to the monomial order <). We note that a normal form is not unique in general.
As a corollary of Lemma 1, we obtain the following:
Proposition 1. 
The set of S-standard monomials spans the algebra A = F X / I defined by the subset S as a vector space over F .
Let p and q be monic polynomials in F X with leading monomials p ¯ and q ¯ . We define the composition of p and q as follows:
Definition 2. 
(a) If there are a and b in X such that p ¯ a = b q ¯ = w with l ( p ¯ ) > l ( b ) , then the composition of intersection is defined as ( p , q ) w = p a b q .
(b) If there are a and b in X such that a 1 , a p ¯ b = q ¯ = w , then the composition of inclusion is defined to be ( p , q ) a , b = a p b q .
Let p , q F X and w X . We define the congruence relation on F X as follows: p q mod ( S ; w ) if and only if p q = α i a i s i b i , where α i F , a i , b i X , s i S , a i s i ¯ b i < w .
Definition 3. 
A subset S of monic elements in F X is said to be closed under composition if ( p , q ) w 0 mod ( S ; w ) and ( p , q ) a , b 0 mod ( S ; w ) for all p , q S , a , b X whenever the compositions ( p , q ) w and ( p , q ) a , b are defined.
The following theorem is the main tool for our results in the next two sections:
Theorem 1 ([6,7]). 
Let S be a subset of monic elements in F X . Then, the following conditions are equivalent:
(a) 
S is closed under composition;
(b) 
For each p F X , a normal form of p with respect to S is unique;
(c) 
The set of S-standard monomials forms the F -linear basis of the algebra A = F X / I defined by S.
Definition 4. 
A subset S of monic elements in F X is a Gröbner–Shirshov basis if S satisfies one of the equivalent conditions in Theorem 1. In this case, we say that S is a Gröbner–Shirshov basis for the algebra A defined by S.
Remark 2. 
The first condition (a) in Theorem 1 gives an analogue of Buchberger’s algorithm or Shirshov’s algorithm. Let S ( 0 ) : = S be any subset of monic elements in F X . For i 0 , set
S ( i ) : = { ( p , q ) w 0 mod ( S ( i ) ; w ) | p , q S ( i ) } { ( p , q ) a , b 0 mod ( S ( i ) ; w ) | p , q S ( i ) } , S ( i + 1 ) : = S ( i ) S ( i ) .
Then, clearly, set S ^ : = i 0 S ( i ) is closed under composition.

3. Temperley–Lieb Algebras of Types A n and B n

3.1. Type A n

First, we review the results ofn the Temperley–Lieb algebra TL ( A n ) . We define TL ( A n ) as the associative algebra over the complex field C , generated by X = { E 1 , E 2 , , E n } with the following defining relations:
  E i 2 = δ E i for   1 i n , ( idempotent   relations ) R TL ( A n ) : E i E j = E j E i for   i > j + 1 , ( commutative   relations ) E i E j E i = E i for   j = i ± 1 , ( untwisting   relations )
where δ C is a parameter. Our monomial order < is taken to be the degree-lexicographic order with
E 1 < E 2 < < E n .
We write E i , j = E i E i 1 E j for i j (hence, E i , i = E i ). By convention, E i , i + 1 = 1 for i 1 .
Proposition 2 
([9], p. 287). The Temperley–Lieb algebra TL ( A n ) has a Gröbner–Shirshov basis as follows:
  E i 2 δ E i for   1 i n , R ^ TL ( A n ) : E i E j E j E i for   i > j + 1 , E i , j E i E i 2 , j E i for   i > j , E j E i , j E j E i , j + 2 for   i > j .
The corresponding R ^ TL ( A n ) -standard monomials are of the form
E i 1 , j 1 E i 2 , j 2 E i p , j p ( 0 p n )
where
1 i 1 < i 2 < < i p n , 1 j 1 < j 2 < < j p n , i 1 j 1 , i 2 j 2 , , i p j p
(the case of p = 0 is the monomial 1). We denote the set of R ^ TL ( A n ) -standard monomials by M TL ( A n ) . Note that the number | M TL ( A n ) | of R ^ TL ( A n ) -standard monomials is
1 n + 2 2 n + 2 n + 1 : = C n + 1 ,
representing the ( n + 1 ) t h Catalan number.
Example 1 
( A 3 ). Note that | M TL ( A 2 ) | = C 3 = 5 and | M TL ( A 3 ) | = C 4 = 14 . Explicitly, the R ^ TL ( A 3 ) -standard monomials are as follows:
1 , E 1 , E 2 , E 1 E 2 , E 2 , 1 , E 3 , E 1 E 3 , E 2 E 3 , E 1 E 2 E 3 , E 2 , 1 E 3 , E 3 , 2 , E 1 E 3 , 2 , E 2 , 1 E 3 , 2 , E 3 , 1 .
Remark 3. 
(1) Considering the standard monomials in the Temperley–Lieb algebra, we see that the product of two standard monomials becomes a standard monomial up to a scalar multiple. For example, if we multiply E 1 E 2 by E 2 , 1 E 3 , 2 in the previous example, then we obtain
( E 1 E 2 ) ( E 2 , 1 E 3 , 2 ) = δ E 1 E 2 E 1 E 3 , 2 = δ E 1 E 3 , 2 ,
which is a multiple of another standard monomial E 1 E 3 , 2 . For another one, the multiplication of E 2 , 1 by E 3 , 1 leads us to have
E 2 , 1 E 3 , 1 = E 2 ( E 1 E 3 , 1 ) = E 2 ( E 1 E 3 ) = E 2 , 1 E 3
based on the Gröbner–Shirshov basis (2).
(2) Note that the collection of sets of pairs ( i 1 , j 1 ) , ( i 2 , j 2 ) , , ( i p , j p ) in (3) is in 1-1 correspondence to the family of paths going to the east and north from ( 0 , 0 ) to ( n + 1 , n + 1 ) in the lattice of integer points, not going above the line of the diagonal points. The cardinality is known as the Catalan number, arising in many counting problems in combinatorics (See [26] for details).
(3) Also, we note that the number of standard monomials is equal to the number of fully commutative elements [19,20].

3.2. Type B 3

Instead of defining the Temperley–Lieb algebra TL ( B n ) ( n 2 ) with
dim TL ( B n ) = ( n + 2 ) C n 1 ,
we just define the algebra of type B 3 in this paper.
Consider the Coxeter diagram for type B 3 as follows:
Symmetry 16 01458 i001
The algebra TL ( B 3 ) is the associative algebra over C , generated by X = { E 1 , E 2 , E 3 } with the following defining relations:
  E i 2 = δ E i for   1 i 3 , R TL ( B 3 ) : E 3 E 1 = E 1 E 3 ,   E i E j E i = E i for   { i , j } = { 1 , 2 } , E i E j E i E j = 2 E i E j for   { i , j } = { 2 , 3 } ,
where δ C is a parameter. We then take our monomial order < to be the degree-lexicographic order with
E 1 < E 2 < E 3 .
Proposition 3 
([24]). The algebra TL ( B 3 ) has a Gröbner–Shirshov basis R ^ TL ( B 3 ) with respect to our monomial order < as follows:
  E i 2 δ E i for   1 i 3 , E 3 E 1 E 1 E 3 , R ^ TL ( B 3 ) : E i E j E i E i for   { i , j } = { 1 , 2 } , E 1 E 3 , 1 E 1 E 3 , E i E j E i E j 2 E i E j for   { i , j } = { 2 , 3 } .
The cardinality of the set M TL ( B 3 ) , i.e., the set of R ^ TL ( B 3 ) -standard monomials, is dim TL ( B 3 ) = ( 3 + 2 ) C 3 1 = 24 .
Explicitly, the R ^ TL ( B 3 ) -standard monomials that are not R ^ TL ( A 3 ) -standard are as follows:
E 2 E 3 , 2 , E 1 E 2 E 3 , 2 , E 2 E 3 , 1 , E 1 E 2 E 3 , 1 , E 3 , 2 E 3 , E 1 E 3 , 2 E 3 , E 2 , 1 E 3 , 2 E 3 , E 3 , 1 E 3 , E 3 , 1 E 3 , 2 , E 3 , 1 E 3 , 2 E 3 .

4. Temperley–Lieb Algebras of Type F n

Now, we consider the Coxeter diagram for type F n :
Symmetry 16 01458 i002
Let TL ( F n ) ( n 4 ) be the Temperley–Lieb algebra of type F n , that is, the associative algebra over C , generated by X = { E 1 , E 2 , , E n } with defining relations as follows:
  E i 2 = δ E i for   1 i n , R TL ( F n ) : E i E j = E j E i for   i > j + 1 , E i E j E i = E i for   { i , j } = { 1 , 2 } , { 3 , 4 } , { 4 , 5 } , , { n 1 , n } , E i E j E i E j = 2 E i E j for   { i , j } = { 2 , 3 } ,
where δ C is a parameter. We then ix our monomial order < to be the degree-lexicographic order with
E 1 < E 2 < < E n .

4.1. Type F 4

From the relations given in R TL ( F 4 ) , we calculate the Gröbner–Shirshov basis for the algebra TL ( F 4 ) .
Theorem 2. 
The algebra TL ( F 4 ) has a Gröbner–Shirshov basis R ^ TL ( F 4 ) with respect to our monomial order < as follows:
  E i 2 δ E i for   1 i 4 , E i E j E j E i for   i > j + 1 , E i E j E i E i for   { i , j } = { 1 , 2 } , { 3 , 4 } , R ^ TL ( F 4 ) : E 1 E i , 1 E 1 E i , 3 for   i > 1 , E 4 , j E 4 E 2 , j E 4 for   j < 3 , E i E j E i E j 2 E i E j for   { i , j } = { 2 , 3 } , E 2 E 4 , j E 3 2 E 2 , j E 4 , 3 for   j < 3 .
The set of R ^ TL ( F 4 ) -standard monomials is denoted by M TL ( F 4 ) . Note that the number | M TL ( F 4 ) | is dim TL ( F 4 ) = 106 (See [27], p. 319).
Explicitly, the R ^ TL ( F 4 ) -standard monomials that are not R ^ TL ( B 3 ) -standard are as follows:
m E 4 ( m M TL ( B 3 ) ) , E 4 , 3 , m E 4 , 3 ( m M TL ( B 3 ) , w h i c h   e n d s   w i t h   E 1   o r   E 2 ) , E 4 , 2 , m E 4 , 2 ( m M TL ( B 3 ) , w h i c h   e n d s   w i t h   E 1   o r   E 2 ) , E 4 , 1 , m E 4 , 1 ( m M TL ( B 3 ) , w h i c h   e n d s   w i t h   E 2 ) , E 4 , 2 E 3 , m E 4 , 2 E 3 ( m M TL ( B 3 ) , w h i c h   e n d s   w i t h   E 1 ) , E 4 , 2 E 3 E 4 , m E 4 , 2 E 3 E 4 ( m M TL ( B 3 ) , w h i c h   e n d s   w i t h   E 1 ) , E 4 , 1 E 3 , E 4 , 1 E 3 E 4 , E 4 , 1 E 3 , 2 , E 4 , 1 E 3 , 2 E 4 , E 4 , 1 E 3 , 2 E 4 , 3 , E 4 , 1 E 3 , 2 E 4 , 2 , E 4 , 1 E 3 , 2 E 4 , 1 , E 4 , 1 E 3 , 2 E 3 , E 4 , 1 E 3 , 2 E 3 E 4 .
Proof. 
First, it can be easily proved that the relations in R ^ TL ( F 4 ) hold. Counting the number of R ^ TL ( F 4 ) -standard monomials, we see that the number | M TL ( F 4 ) | = 106 is exactly the dimension of the Temperley–Lieb algebra TL ( F 4 ) , which is the same as the number of fully commutative elements of type F 4 , as given in [27]. Hence, by Theorem 1 (c), the set R ^ TL ( F 4 ) is the Gröbner–Shirshov basis for the algebra TL ( F 4 ) defined by R TL ( F 4 ) . □
Example 2. 
We have the multiplication table between the standard monomials. For example, we multiply the monomial E 4 , 1 E 3 , 2 E 3 E 4 by E 1 from the right to obtain
E 4 , 1 E 3 , 2 E 3 E 4 E 1 = E 4 , 1 E 3 , 1 E 3 E 4 = E 4 , 2 E 1 E 3 E 3 E 4 = δ E 4 , 1 E 3 E 4
from the relations in R ^ TL ( F 4 ) .

4.2. Type F 5

Adding one more generator, we compute the Gröbner–Shirshov and the corresponding finite set of standard monomials.
Theorem 3. 
The algebra TL ( F 5 ) has a Gröbner–Shirshov basis R ^ TL ( F 5 ) with respect to our monomial order < as follows:
  E i 2 δ E i for   1 i 5 , E i E j E j E i for   i > j + 1 , E i E j E i E i for   { i , j } = { 1 , 2 } , { 3 , 4 } , { 4 , 5 } E 1 E i , 1 E 1 E i , 3 for   i > 1 , R ^ TL ( F 5 ) : E 3 E 5 , 3 E 3 E 5 , E i , j E i E i 2 , j E i for   i > 3 j , E 5 , 2 E 3 E 5 E 3 , 2 E 3 E 5 , E i E j E i E j 2 E i E j for   { i , j } = { 2 , 3 } , E 2 E i , j E 3 2 E 2 , j E i , 3 for   i > 3 > j .
We denote the set of R ^ TL ( F 5 ) -standard monomials based on M TL ( F 5 ) . Note that the number | M TL ( F 5 ) | is dim TL ( F 5 ) = 464 (See [27], p. 319).
Explicitly, the R ^ TL ( F 5 ) -standard monomials that are not R ^ TL ( F 4 ) -standard are as follows:
m E 5 ( m M TL ( F 4 ) ) , E 5 , 4 , m E 5 , 4 ( m M TL ( F 4 ) , w h i c h   e n d s   w i t h   E 1   o r   E 2   o r   E 3 ) , E 5 , 3 , m E 5 , 3 ( m M TL ( F 4 ) , w h i c h   e n d s   w i t h   E 1   o r   E 2 ) , E 5 , 2 , m E 5 , 2 ( m M TL ( F 4 ) , w h i c h   e n d s   w i t h   E 1   o r   E 2 ) , E 5 , 1 , m E 5 , 1 ( m M TL ( F 4 ) , w h i c h   e n d s   w i t h   E 2 ) , E 5 , 2 E 3 , m E 5 , 2 E 3 ( m M TL ( F 4 ) , w h i c h   e n d s   w i t h   E 1 ) , E 5 , 2 E 3 E 4 , m E 5 , 2 E 3 E 4 ( m M TL ( F 4 ) , w h i c h   e n d s   w i t h   E 1 ) , E 5 , 2 E 3 E 4 E 5 , m E 5 , 2 E 3 E 4 E 5 ( m M TL ( F 4 ) , w h i c h   e n d s   w i t h   E 1 ) , E 5 , 1 E 3 , E 5 , 1 E 3 E 4 , E 5 , 1 E 3 , 2 , E 5 , 1 E 3 , 2 E 4 , E 5 , 1 E 3 , 2 E 4 , 3 , E 5 , 1 E 3 , 2 E 4 , 2 , E 5 , 1 E 3 , 2 E 4 , 1 , E 5 , 1 E 3 , 2 E 3 , E 5 , 1 E 3 , 2 E 3 E 4 , E 5 , 1 E 3 E 4 E 5 , E 5 , 1 E 3 , 2 E 4 E 5 , E 5 , 1 E 3 , 2 E 4 , 3 E 5 , E 5 , 1 E 3 , 2 E 4 , 3 E 5 , 4 , E 5 , 1 E 3 , 2 E 4 , 2 E 5 , E 5 , 1 E 3 , 2 E 4 , 2 E 5 , 4 , E 5 , 1 E 3 , 2 E 4 , 2 E 5 , 3 , E 5 , 1 E 3 , 2 E 4 , 2 E 5 , 2 , E 5 , 1 E 3 , 2 E 4 , 2 E 5 , 1 , E 5 , 1 E 3 , 2 E 4 , 1 E 5 , E 5 , 1 E 3 , 2 E 4 , 1 E 5 , 4 , E 5 , 1 E 3 , 2 E 4 , 1 E 5 , 3 , E 5 , 1 E 3 , 2 E 4 , 1 E 5 , 2 , E 5 , 1 E 3 , 2 E 4 , 1 E 5 , 2 E 3 , E 5 , 1 E 3 , 2 E 4 , 1 E 5 , 2 E 3 E 4 , E 5 , 1 E 3 , 2 E 4 , 1 E 5 , 2 E 3 E 4 E 5 , E 5 , 1 E 3 , 2 E 3 E 4 E 5 .

4.3. Type F 6

Theorem 4. 
The algebra TL ( F 6 ) has a Gröbner–Shirshov basis R ^ TL ( F 6 ) with respect to our monomial order < as follows:
  E i 2 δ E i for   1 i 6 , E i E j E j E i for   i > j + 1 , E i E j E i E i for   { i , j } = { 1 , 2 } , { 3 , 4 } , { 4 , 5 } , { 5 , 6 } E 1 E i , 1 E 1 E i , 3 for   i > 1 , R ^ TL ( F 5 ) : E j E i , j E j E i , j + 2 for   i > j + 1 > 3 , E i , j E i E i 2 , j E i for   i > 3 j , i > j > 3 E i , 2 E 3 E i E i 2 , 2 E 3 E i for   i > 4 , E i E j E i E j 2 E i E j for   { i , j } = { 2 , 3 } , E 2 E i , j E 3 2 E 2 , j E i , 3 for   i > 3 > j .
We denote the set of R ^ TL ( F 6 ) -standard monomials by M TL ( F 6 ) . Note that the number | M TL ( F 6 ) | is dim TL ( F 5 ) = 2003 (See [27], p. 319).
Explicitly, the R ^ TL ( F 6 ) -standard monomials that are not R ^ TL ( F 5 ) -standard are as follows:
m E 6 ( m M TL ( F 5 ) ) , E 6 , 5 , m E 6 , 5 ( m M TL ( F 5 ) , w h i c h   e n d s   w i t h   E i   ( i < 5 ) , E 6 , 4 , m E 6 , 4 ( m M TL ( F 5 ) , w h i c h   e n d s   w i t h   E i   ( i < 4 ) , E 6 , 3 , m E 6 , 3 ( m M TL ( F 5 ) , w h i c h   e n d s   w i t h   E i   ( i < 3 ) , E 6 , 2 , m E 6 , 2 ( m M TL ( F 5 ) , w h i c h   e n d s   w i t h   E i   ( i < 3 ) , E 6 , 1 , m E 6 , 1 ( m M TL ( F 5 ) , w h i c h   e n d s   w i t h   E 2 ) , E 6 , 2 E 3 , m E 6 , 2 E 3 ( m M TL ( F 5 ) , w h i c h   e n d s   w i t h   E 1 ) , E 6 , 2 E 3 E 4 , m E 6 , 2 E 3 E 4 ( m M TL ( F 5 ) , w h i c h   e n d s   w i t h   E 1 ) , E 6 , 2 E 3 E 4 E 5 , m E 6 , 2 E 3 E 4 E 5 ( m M TL ( F 5 ) , w h i c h   e n d s   w i t h   E 1 ) , E 6 , 2 E 3 E 4 E 5 E 6 , m E 6 , 2 E 3 E 4 E 5 E 6 ( m M TL ( F 5 ) , w h i c h   e n d s   w i t h   E 1 ) ,
E 6 , 1 E 3 , E 6 , 1 E 3 E 4 , E 6 , 1 E 3 , 2 , E 6 , 1 E 3 , 2 E 4 , E 6 , 1 E 3 , 2 E 4 , 3 , E 6 , 1 E 3 , 2 E 4 , 2 , E 6 , 1 E 3 , 2 E 4 , 1 , E 6 , 1 E 3 , 2 E 3 , E 6 , 1 E 3 , 2 E 3 E 4 , E 6 , 1 E 3 E 4 E 5 , E 6 , 1 E 3 , 2 E 4 E 5 , E 6 , 1 E 3 , 2 E 4 , 3 E 5 , E 6 , 1 E 3 , 2 E 4 , 3 E 5 , 4 , E 6 , 1 E 3 , 2 E 4 , 2 E 5 , E 6 , 1 E 3 , 2 E 4 , 2 E 5 , 4 , E 6 , 1 E 3 , 2 E 4 , 2 E 5 , 3 , E 6 , 1 E 3 , 2 E 4 , 2 E 5 , 2 , E 6 , 1 E 3 , 2 E 4 , 2 E 5 , 1 , E 6 , 1 E 3 , 2 E 4 , 1 E 5 , E 6 , 1 E 3 , 2 E 4 , 1 E 5 , 4 , E 6 , 1 E 3 , 2 E 4 , 1 E 5 , 3 , E 6 , 1 E 3 , 2 E 4 , 1 E 5 , 2 , E 6 , 1 E 3 , 2 E 4 , 1 E 5 , 2 E 3 , E 6 , 1 E 3 , 2 E 4 , 1 E 5 , 2 E 3 E 4 , E 6 , 1 E 3 , 2 E 4 , 1 E 5 , 2 E 3 E 4 E 5 , E 6 , 1 E 3 , 2 E 3 E 4 E 5 , E 6 , 1 E 3 E 4 E 5 E 6 , E 6 , 1 E 3 , 2 E 4 E 5 E 6 , E 6 , 1 E 3 , 2 E 4 , 3 E 5 E 6 , E 6 , 1 E 3 , 2 E 4 , 3 E 5 , 4 E 6 , E 6 , 1 E 3 , 2 E 4 , 3 E 5 , 4 E 6 , 5 , E 6 , 1 E 3 , 2 E 4 , 2 E 5 E 6 , E 6 , 1 E 3 , 2 E 4 , 2 E 5 , 4 E 6 , E 6 , 1 E 3 , 2 E 4 , 2 E 5 , 4 E 6 , 5 , E 6 , 1 E 3 , 2 E 4 , 2 E 5 , 3 E 6 , E 6 , 1 E 3 , 2 E 4 , 2 E 5 , 3 E 6 , 5 , E 6 , 1 E 3 , 2 E 4 , 2 E 5 , 3 E 6 , 4 , E 6 , 1 E 3 , 2 E 4 , 2 E 5 , 2 E 6 , E 6 , 1 E 3 , 2 E 4 , 2 E 5 , 2 E 6 , 5 , E 6 , 1 E 3 , 2 E 4 , 2 E 5 , 2 E 6 , 4 , E 6 , 1 E 3 , 2 E 4 , 2 E 5 , 2 E 6 , 3 , E 6 , 1 E 3 , 2 E 4 , 2 E 5 , 2 E 6 , 2 , E 6 , 1 E 3 , 2 E 4 , 2 E 5 , 2 E 6 , 1 , E 6 , 1 E 3 , 2 E 4 , 2 E 5 , 1 E 6 , E 6 , 1 E 3 , 2 E 4 , 2 E 5 , 1 E 6 , 5 , E 6 , 1 E 3 , 2 E 4 , 2 E 5 , 1 E 6 , 4 , E 6 , 1 E 3 , 2 E 4 , 2 E 5 , 1 E 6 , 3 , E 6 , 1 E 3 , 2 E 4 , 2 E 5 , 1 E 6 , 2 , E 6 , 1 E 3 , 2 E 4 , 2 E 5 , 1 E 6 , 2 E 3 , E 6 , 1 E 3 , 2 E 4 , 2 E 5 , 1 E 6 , 2 E 3 E 4 , E 6 , 1 E 3 , 2 E 4 , 2 E 5 , 1 E 6 , 2 E 3 E 4 E 5 , E 6 , 1 E 3 , 2 E 4 , 2 E 5 , 1 E 6 , 2 E 3 E 4 E 5 E 6 , E 6 , 1 E 3 , 2 E 4 , 1 E 5 E 6 , E 6 , 1 E 3 , 2 E 4 , 1 E 5 , 4 E 6 , E 6 , 1 E 3 , 2 E 4 , 1 E 5 , 4 E 6 , 5 , E 6 , 1 E 3 , 2 E 4 , 1 E 5 , 3 E 6 , E 6 , 1 E 3 , 2 E 4 , 1 E 5 , 3 E 6 , 5 , E 6 , 1 E 3 , 2 E 4 , 1 E 5 , 3 E 6 , 4 , E 6 , 1 E 3 , 2 E 4 , 1 E 5 , 2 E 6 , E 6 , 1 E 3 , 2 E 4 , 1 E 5 , 2 E 6 , 5 , E 6 , 1 E 3 , 2 E 4 , 1 E 5 , 2 E 6 , 4 , E 6 , 1 E 3 , 2 E 4 , 1 E 5 , 2 E 6 , 3 , E 6 , 1 E 3 , 2 E 4 , 1 E 5 , 2 E 6 , 2 , E 6 , 1 E 3 , 2 E 4 , 1 E 5 , 2 E 6 , 1 , E 6 , 1 E 3 , 2 E 4 , 1 E 5 , 2 E 3 E 6 , E 6 , 1 E 3 , 2 E 4 , 1 E 5 , 2 E 3 E 6 , 5 , E 6 , 1 E 3 , 2 E 4 , 1 E 5 , 2 E 3 E 6 , 4 , E 6 , 1 E 3 , 2 E 4 , 1 E 5 , 2 E 3 E 4 E 6 , E 6 , 1 E 3 , 2 E 4 , 1 E 5 , 2 E 3 E 4 E 6 , 5 , E 6 , 1 E 3 , 2 E 4 , 1 E 5 , 2 E 3 E 4 E 5 E 6 , E 6 , 1 E 3 , 2 E 3 E 4 E 5 E 6 .

4.4. Type F n ( n 7 )

In general, we obtain a Gröbner–Shirshov for the Temperley–Lieb algebra of type F n ( n 7 ) as follows:
Theorem 5. 
The algebra TL ( F n ) has a Gröbner–Shirshov basis R ^ TL ( F n ) with respect to our monomial order < as follows:
  E i 2 δ E i for   1 i n , E i E j E j E i for   i > j + 1 , E i E j E i E i for   { i , j } = { 1 , 2 } , { 3 , 4 } , { 4 , 5 } , { 5 , 6 } , , { n 1 , n } E 1 E i , 1 E 1 E i , 3 for   i > 1 , R ^ TL ( F n ) : E j E i , j E j E i , j + 2 for   i > j + 1 > 3 , E i , j E i E i 2 , j E i for   i > 3 j , i > j > 3 E i , 2 E 3 E i E i 2 , 2 E 3 E i for   i > 4 , E i E j E i E j 2 E i E j for   { i , j } = { 2 , 3 } , E 2 E i , j E 3 2 E 2 , j E i , 3 for   i > 3 > j .
We denote the set of R ^ TL ( F n ) -standard monomials with M TL ( F n ) . Note that the number | M TL ( F n ) | is
dim TL ( F n ) = 5 f 3 n 4 5 k = 2 n 1 f 3 k 5 n k + 1 2 n 2 k n k + 1 n 2 n 2 n 1 2 n f 2 n 2 2 f 2 n 4 + f n 1 1
where { f n } is the Fibonacci sequence (See [27], p. 306).
Proof. 
We use the algorithm following part (a) of Theorem 1. Taking all possible compositions from the relations in the previous step, we obtain new relations that cannot be reducible with respect to the prior relations. It is clearly shown that all the relations in R ^ TL ( F n ) hold and that any composition between the polynomials in R ^ TL ( F n ) reduces to 0 with respect to R ^ TL ( F n ) , which means that R ^ TL ( F n ) is closed under composition. □
Remark 4. 
For the Temperley–Lieb algebras of types E n ( n 6 ) and H n ( n 2 ) , we are working on their Gröbner–Shirshov bases and the corresponding standard monomials, which will be a new interpretation for the fully commutative elements of the associated types.

Author Contributions

Conceptualization, J.-Y.L. and D.-i.L.; methodology, J.-Y.L. and D.-i.L.; formal analysis, J.-Y.L. and D.-i.L.; investigation, J.-Y.L. and D.-i.L.; writing—original draft preparation, J.-Y.L. and D.-i.L.; writing—review and editing, J.-Y.L. and D.-i.L.; project administration, J.-Y.L. and D.-i.L.; funding acquisition, J.-Y.L. and D.-i.L. All authors have read and agreed to the published version of the manuscript.

Funding

The work of the first author was supported by a National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIT) (No. RS-2023-00251466), and the corresponding author’s research was supported by NRF Grant # 2018R1D1A1B07044111 and a research grant from Seoul Women’s University (2024-0254).

Data Availability Statement

Data are contained within the article.

Conflicts of Interest

The authors declare no conflicts of interest.

References

  1. Temperley, H.N.V.; Lieb, E.H. Relations between percolation and colouring problems and other graph theoretical problems associated with regular planar lattices: Some exact results for the percolation problem. Proc. R. Soc. Lond. Ser. A 1971, 322, 251–280. [Google Scholar]
  2. Jones, V.F.R. Hecke algebra representations of braid groups and link polynomials. Ann. Math. 1987, 126, 335–388. [Google Scholar] [CrossRef]
  3. Buchberger, B. An Algorithm for Finding the Basis Elements of the Residue Class Ring of a Zero Dimensional Polynomial Ideal. Ph.D. Thesis, University of Innsbruck, Innsbruck, Austria, 1965. (In German) Translated in J. Symbolic Comput. 2006, 41, 475–511. [Google Scholar]
  4. Shirshov, A.I. Some algorithmic problems for Lie algebras. Sibirk. Math. Z. 1962, 3, 292–296. (In Russian) Translated in ACM SIGSAM Bull. Commun. Comput. Algebra 1999, 33, 3–6 [Google Scholar]
  5. Shirshov, A.I. Selected Works of AI Shirshov; Bokut, L.A., Latyshev, V., Shestakov, I., Zelmanov, E., Eds.; Birkhäuser: Basel, Switzerland, 2009. [Google Scholar]
  6. Bokut, L.A. Imbedding into simple associative algebras. Algebra Logic 1976, 15, 117–142. [Google Scholar] [CrossRef]
  7. Bergman, G.M. The diamond lemma for ring theory. Adv. Math. 1978, 29, 178–218. [Google Scholar] [CrossRef]
  8. Miller, C.F. Decision problems for groups—Survey and reflections. In Algorithms and Classification in Combinatorial Group Theory; Baumslag, G., Miller, C.F., Eds.; Springer: Berlin/Heidelberg, Germany, 1992; Volume 23, pp. 1–59. [Google Scholar]
  9. Kang, S.-J.; Lee, I.-S.; Lee, K.-H.; Oh, H. Hecke algebras, Specht modules and Gröbner-Shirshov bases. J. Algebra 2002, 252, 258–292. [Google Scholar] [CrossRef]
  10. Bokut, L.A.; Shiao, L.-S. Gröbner-Shirshov bases for Coxeter groups. Comm. Algebra 2001, 29, 4305–4319. [Google Scholar] [CrossRef]
  11. Borges-Trenard, M.A.; Pérez-Rosés, H. Complete presentations of direct products of groups. Cien. Mat. 2001, 19, 3–11. [Google Scholar]
  12. Lee, D.-I. Standard monomials for the Weyl group F4. J. Algebra Appl. 2016, 15, 1650146. [Google Scholar] [CrossRef]
  13. Lee, D. Gröbner-Shirshov bases and normal forms for the Coxeter groups E6 and E7. In Advances in Algebra and Combinatorics; Shum, K.P., Zelmanov, E., Zhang, Z., Shangzhi, L., Eds.; World Scientific Publications: Singapore, 2008; pp. 243–255. [Google Scholar]
  14. Svechkarenko, O. Gröbner-Shirshov Bases for the Coxeter Group E8. Master’s Thesis, Novosibirsk State University, Novosibirsk, Russia, 2007. [Google Scholar]
  15. Lee, J.-Y.; Lee, D.-I. Gröbner-Shirshov bases for non-crystallographic Coxeter groups. Acta Cryst. 2019, A75, 584–592. [Google Scholar] [CrossRef] [PubMed]
  16. Bokut, L.A.; Chen, Y. Gröbner-Shirshov bases and PBW theorems. J. Sib. Fed. Univ. Math. Phys. 2013, 6, 417–427. [Google Scholar]
  17. Ivanov-Pogodaev, I.; Malcev, S. Finite Gröbner basis algebras with unsolvable nilpotency problem and zero divisors problem. J. Algebra 2018, 508, 575–588. [Google Scholar] [CrossRef]
  18. Oner, T.; Senturk, I.; Oner, G. An independent set of axioms of MV-algebras and solutions of the set-theoretical Yang–Baxter equation. Axioms 2017, 6, 17. [Google Scholar] [CrossRef]
  19. Graham, J.J. Modular Representations of Hecke Algebras and Related Algebras. Ph.D. Thesis, University of Sydney, Sydney, Australia, 1995. [Google Scholar]
  20. Fan, C.K. Structure of a Hecke algebra quotient. J. Am. Math. Soc. 1997, 10, 139–167. [Google Scholar] [CrossRef]
  21. Kleshchev, A.; Ram, A. Homogeneous representations of Khovanov-Lauda algebras. J. Eur. Math. Soc. 2010, 12, 1293–1306. [Google Scholar] [CrossRef]
  22. Feinberg, G.; Lee, K.-H. Fully commutative elements of type D and homogeneous representations of KLR-algebras. J. Comb. 2015, 6, 535–557. [Google Scholar] [CrossRef]
  23. Stembridge, J.R. On the fully commutative elements of Coxeter groups. J. Algebr. Comb. 1996, 5, 353–385. [Google Scholar] [CrossRef]
  24. Kim, S.; Lee, D.-I. Gröbner-Shirshov bases for Temperley-Lieb algebras of types B and D. J. Algebra Appl. 2020, 19, 2050002. [Google Scholar] [CrossRef]
  25. Lee, J.-Y.; Lee, D.-I.; Kim, S. Gröbner-Shirshov bases for Temperley-Lieb algebras of complex reflection groups. Symmetry 2018, 10, 438. [Google Scholar] [CrossRef]
  26. Stanley, R.P. Catalan Numbers; Cambridge University Press: Cambridge, UK, 2015. [Google Scholar]
  27. Stembridge, J.R. The enumeration of fully commutative elements of Coxeter groups. J. Algebr. Comb. 1998, 7, 291–320. [Google Scholar] [CrossRef]
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Share and Cite

MDPI and ACS Style

Lee, J.-Y.; Lee, D.-i. Gröbner–Shirshov Bases for Temperley–Lieb Algebras of Type F. Symmetry 2024, 16, 1458. https://doi.org/10.3390/sym16111458

AMA Style

Lee J-Y, Lee D-i. Gröbner–Shirshov Bases for Temperley–Lieb Algebras of Type F. Symmetry. 2024; 16(11):1458. https://doi.org/10.3390/sym16111458

Chicago/Turabian Style

Lee, Jeong-Yup, and Dong-il Lee. 2024. "Gröbner–Shirshov Bases for Temperley–Lieb Algebras of Type F" Symmetry 16, no. 11: 1458. https://doi.org/10.3390/sym16111458

APA Style

Lee, J.-Y., & Lee, D.-i. (2024). Gröbner–Shirshov Bases for Temperley–Lieb Algebras of Type F. Symmetry, 16(11), 1458. https://doi.org/10.3390/sym16111458

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

Article Metrics

Back to TopTop